Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 265
m
1.0
0.5
0.0
–0.5
T < T c
T = T c
T > T c
–1.0
–2.0 –1.0 0.0 1.0 2.0
H
Figure 5.15: The equilibrium values of m as a function of the magnetic field H for T > Tc, T = Tc,
and T < Tc. The plot for T > Tc is smooth in contrast to the plot for T < Tc which has a
discontinuity at H = 0. For T = Tc there is no discontinuity, and the function m(H) has an
infinite slope at H = 0.
characteristic of discontinuous phase transitions. An example of the data you can obtain in
this way is shown in Figure 5.16.
(d) Change the number of mcs per field value to 1 and view the resulting plot for m versus H.
Repeat for mcs per field value equal to 100. Explain the differences you see.
5.8 *Simulation of the Density of States
The probability that a system in equilibrium with a heat bath at a temperature T has energy E
is given by
P(E) = Ω(E)
Z e−βE , (5.129)
whereΩ(E)isthenumberofstateswithenergyE, 11 andthepartitionfunctionZ =
EΩ(E)e−βE .
If Ω(E) is known, we can calculate the mean energy (and other thermodynamic quantities) at any
temperature from the relation
E = 1
Z
EΩ(E)e −βE . (5.130)
E
Hence, the quantityΩ(E) is of much interest. In the followingwe discuss an algorithmthat directly
computes Ω(E) for the Ising model. In this case the energy is a discrete variable and hence the
quantity we wish to compute is the number of spin microstates with the same energy.
11 The quantity Ω(E) is the number of states with energy E for a system such as the Ising model which has discrete
values of the energy. It is common to refer to Ω(E) as the density of states even when the values of E are discrete.